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saying that the sum of the reciprocals of numbers in some infinite set converges is essentially saying that the numbers get bigger faster than they get numerous
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this is just a random realization i got while watching a numberphile video that mentions how the sum of the reciprocals of all numbers that don't contain a 7 in their decimal expansion (or that don't contain a 0, or a 1, etc) converges, which means they're less numerous than the primes since the sum of the reciprocals of the primes diverges
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